Niels Henrik Abel

Niels Henrik Abel was a Norwegian mathematician who, at the age of twenty-one, proved that there is no general algebraic formula, using only addition, subtraction, multiplication, division and radicals such as square and cube roots, that can solve every fifth-degree polynomial equation, a result now known as the Abel-Ruffini theorem that finally settled a problem mathematicians had pursued since the sixteenth century, when similar formulas had been found for equations of degree four and lower. Abel's proof, published in 1824 at his own expense because he could not afford wider printing, grew out of methods that also helped lay foundations for what became group theory, and the important class of groups now called abelian groups, in which the order of combining elements does not affect the result, are named in his honor. He also did extensive foundational work on elliptic functions and elliptic integrals, developing results independently of and roughly contemporaneously with Carl Gustav Jacobi that became central to nineteenth-century analysis. Abel struggled throughout his short career to gain recognition and a stable academic position despite the significance of his work, and he died of tuberculosis in 1829 at the age of twenty-six, just before a letter arrived offering him a professorship in Berlin; the Abel Prize, one of the most prestigious international awards in mathematics, was established by Norway in 2001 in his memory.